We study the stability of a radial liquid sheet produced by head-on impingement of two equal laminar liquid jets. Linear stability equations are derived from the inviscid flow equations for a radially expanding sheet that govern the time-dependent evolution of the two liquid interfaces. The analysis accounts for the varying liquid sheet thickness while the inertial effects due to the surrounding gas phase are ignored. The analysis results in stability equations for the sinuous and the varicose modes of sheet deformation that are decoupled at the lowest order of approximation. When the sheet is excited at a fixed frequency, a small sinuous displacement introduced at the point of impingement grows as it is convected downstream suggesting that the sheet is unstable at all Weber numbers (We ≡ ρlU2h/σ) in the absence of the gas phase. Here, ρl is the density of the liquid, U is the speed of the liquid jet, h is the local sheet thickness, and σ is the surface tension. The sinuous disturbance diverges at We = 2 which sets the size of the sheet, in agreement with the results of Taylor [“The dynamics of thin sheets of fluid. III. Disintegration of fluid sheets,” Proc. R. Soc. London, Ser. A 253, 313 (1959)]. Asymptotic analysis of the sinuous mode for all frequencies shows that the disturbance amplitude diverges inversely with the distance from the edge of the sheet. The varicose waves, on the other hand, are neutrally stable at all frequencies and are convected at the speed of the liquid jet.

A recent theory [Tirumkudulu & Paramati, "Stability of a moving radial liquid sheet: Time dependent equations.", Phys. of Fluids, 102107, 25, 2013] for a radially expanding liquid sheet that accounts for liquid inertia, interfacial tension and thinning of the liquid sheet while ignoring inertia of surrounding gas and viscous effects shows that such a sheet is convectively unstable at all frequencies and Weber numbers (W e ≡ ρ l U 2 h/σ) to small sinuous disturbances. Here, ρ l and σ are the density and surface tension of the liquid, respectively, U is the speed of the liquid jet, and h is the local sheet thickness. In this study, we use a simple non-contact optical technique based on laser induced fluorescence to measure the instantaneous local sheet thickness and its displacement of a circular sheet produced by head on impingement of two laminar jets. When the impingement point is disturbed via acoustic forcing, sinuous waves produced close to the impingement point travel radially outward. The phase speed of the sinuous wave decreases while the amplitude grows as they propagate radially outwards. Our experimental technique was unable to detect thickness variations in the presence of forcing suggesting that the variations could be smaller than the resolution of our experimental technique. The measured phase speed of the sinuous wave envelope matches with predictions while there is a qualitative agreement in case of spatial growth. We show that there is a range of frequencies over which the sheet is unstable due to both aerodynamic interaction and thinning effects while outside this range, thinning effects dominate. These results imply that a full theory that describes the dynamics of a radially expanding liquid sheet should account for both effects.

A smooth circular moving liquid sheet is formed by the head-on impingement of two equal laminar water jets. We subject such a liquid sheet to uniform laminar air flow from one side such that the direction of air velocity is perpendicular to the liquid sheet. The pressure of the moving air deforms the liquid sheet giving rise to an open water bell. The water bell is symmetric suggesting that the gas flow around the bell is also symmetric and that the gravitational force is negligible. We have captured the shape of the water bells for varying air flow rates and for varying Weber numbers, and compared the measurements with theoretical predictions obtained from a force balance involving liquid inertia, surface tension, and pressure difference across the sheet. The pressure exerted by the gas phase on the front and the rear surface of the deformed liquid sheet is obtained from known results of flow past flat circular discs. The predicted steady state shapes match well with the measurements at low Weber numbers but differences are observed at high Weber numbers, where the sheet flaps and is no longer smooth. Interestingly, the shape predicted by assuming a constant pressure difference equal to the stagnation pressure over the whole of the front face of the sheet and free stream value over the whole of the rear face yields nearly identical results suggesting that an open water bell is similar to a closed water bell in that, to a good approximation, the pressure on either sides of the water bell is homogeneous.

Sulfur removal is one of the key functions of vacuum gas oil (VGO) hydrotreating reactors. Knowing feed and product properties real-time or near-real-time improves reactor operations. The VGO section of crude distillation unit is also prone to severe high-temperature sulfidic and naphthenic acid corrosion. In this article, we evaluate a single-reflectance attenuated total reflectance Fourier transform infrared (ATR-FTIR) spectroscopy as a possible quick and cost-effective methodology to determine total carboxylic acid number (TCAN) and total sulfur content of VGO. The study shows that single-reflectance diamond ATR crystal methodology has the right signal-to-noise ratio to accurately predict TCAN and total sulfur within the primary method’s repeatability. Statistical models have been developed using 64 sample sets of vacuum gas oil and out of which 10 samples were used for the cross-validation of the model. The range of TCAN in VGO samples used in this study was between 0.37 and 13.8 mg KOH/g, and sulfur content was between 0.8 to 5.4% by mass. Models have been evaluated by determining the correlation coefficient (R 2), linearity curves obtained by plotting measured versus predicted values, and the errors associated with the prediction and cross-validation. The models showed a correlation coefficient of 0.9991 for TCAN and 0.9974 for total sulfur between reference and the measured values for calibration set of samples. A root-mean-square error of calibration (RMSEC) and prediction (RMSEP) for TCAN were found to be 0.0903 and 0.0885 mg KOH/g. Similarly, RMSEC and RMSEP values for sulfur content were 0.0829 and 0.107% by mass, respectively. The proposed methodology for the prediction of total sulfur and TCAN is fast, efficient, and cost-effective and has several advantages over the standard methods.